I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions Empiricism, Probability, and Knowledge of Arithmetic Sean Walsh Department of Logic and Philosophy of Science University of California, Irvine [email protected] http://www.swalsh108.org Progic 2013 Munich September 18, 2013 I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions Introduction: Thesis to be Examined The thesis that I want to examine today is this: arithmetical knowledge may be legitimately extended by confirmation just as it may be by proof. This thesis is one component of a broader empirical account of arithmetical knowledge, according to which: subsequent to some appropriately empirical acquisition of knowledge of quantifier-free truths like 7 + 5 = 12, one may legitimately extend this knowledge to knowledge of more complex arithmetical truths by way of confirmation. My goal today is to defend this thesis against 2 objections of the form: this kind of probability is too close to arithmetical truth. I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions Background: Probabilistic Confirmation Among notions of confirmation, I focus on probabilistic variants: Hypothesis h is confirmed by evidence e relative to background knowledge K if P(hje & K) > P(hjK) Here P is a probability assignment, a function P : Sent(L) ! R which satisfies, for all '; in the signature L of arithmetic: (P1) P(') ≥ 0, (P2) P(') = 1 if j= ', (P3) P(' _ ) = P(') + P( ) if j= :(' & ) where j= is the usual consequence relation from first-order logic. So we're conceiving of probabilities being assigned to fine-grained entities like sentences as opposed to coarse-grained entities like sets of possible worlds. I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions The Canonical Axioms for Arithmetic Among arithmetical axioms, I focus on the Peano axioms, given by: Robinson's Q, a finite set of axioms that imply all the addition and multiplication tables, e.g. 7 + 5 = 12. It has axioms like 8 x s(x) 6= 0. Mathematical Induction, a principle that articulates a canonical means by which to establish a universal hypothesis about natural numbers. It says: if zero has a property F and n + 1 has F whenever n has F , then all natural numbers have F . Two special cases of the thesis are then: I. Let Robinson's Q be written h ≡ 8 x F (x) and consider evidence VN of form e ≡ i=1 F (ai ). Then one may justifiably infer from e to h on the basis of justification in e and h's being confirmed by e. II. Let h ≡ 8 x G(x) and e ≡ [G(0) & 8 n (G(n) ! G(n + 1))]. Then one may justifiably infer from e to h against K ≡ Robinson's Q on the basis of justification in e; K and h's being confirmed by e relative to the background of K. I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions Two Special Cases of the Thesis The thesis that I want to examine today is this: arithmetical knowledge may be legitimately extended by confirmation just as it may be by proof. Among notions of confirmation, I focus on probabilistic variants: Hypothesis h is confirmed by evidence e relative to background knowledge K if P(hje & K) > P(hjK) I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions Two Special Cases of the Thesis The thesis that I want to examine today is this: arithmetical knowledge may be legitimately extended by confirmation just as it may be by proof. Among notions of confirmation, I focus on probabilistic variants: Hypothesis h is confirmed by evidence e relative to background knowledge K if P(hje & K) > P(hjK) Two special cases of the thesis are then: I. Let Robinson's Q be written h ≡ 8 x F (x) and consider evidence VN of form e ≡ i=1 F (ai ). Then one may justifiably infer from e to h on the basis of justification in e and h's being confirmed by e. II. Let h ≡ 8 x G(x) and e ≡ [G(0) & 8 n (G(n) ! G(n + 1))]. Then one may justifiably infer from e to h against K ≡ Robinson's Q on the basis of justification in e; K and h's being confirmed by e relative to the background of K. I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions Significance of the Thesis Two motivating thoughts on the significance of the thesis: First, it accounts for our knowledge of arithmetical axioms, using a source of justification which is routinely and efficaciously employed in other parts of our ordinary reasoning, thus serving to partially dissipate a skeptical concern about the possibility of mathematical knowledge. Second, the parallel between mathematical knowledge and scientific knowledge (broadly construed) is widely accepted in many parts of philosophy of mathematics. However, the one place in philosophy of mathematics that has traditionally resisted any form of empirical treatment is arithmetic{ here, by contrast, logicist and fictionalist approaches have been dominant. I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions The Approach Requires Specification of Probability Assignments, The approach that I adopt obviously requires the specification of some class of probability assignments. Today, I want to focus on two objections, which have the form: this kind of probability assignment, and hence this kind of confirmation, is too close to arithmetical truth. If there was such conceptual proximity, then: any claim to this general approach being broadly empirical in character would be severely undermined, and any skeptical doubts that we could come to be justified in believing arithmetical truths would be equally strong doubts that we could come to be justified in believing that such-and-such sentences had such-and-such probability. I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions Outline I. Introduction II. Avoiding Counting Assignments III. Avoiding Alignment of True and Probable IV. Summary plus Further Questions The philosophical concern relates to true arithmetic Th(N), the set of all sentences true on the standard model of arithmetic. If T1;:::; Tn included true arithmetic Th(N), then one would want to know what it is about our relationship to true arithmetic that grants it this preferred status. While if not, then one would want to know what about T1;:::; Tn make them reliable indicators of truth. Why these and not others? . This dilemma suggests that we should avoid appeal to counting assignments . But how can we do that? I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions The Problem So one kind of probability assignment is a counting assignment: P(') = a1 · T1(') + ··· + an · Tn(') wherein a1 + ··· + an = 1 and T1;:::; Tn are in [K], the space of complete consistent extensions of our background knowledge K. If T1;:::; Tn included true arithmetic Th(N), then one would want to know what it is about our relationship to true arithmetic that grants it this preferred status. While if not, then one would want to know what about T1;:::; Tn make them reliable indicators of truth. Why these and not others? . This dilemma suggests that we should avoid appeal to counting assignments . But how can we do that? I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions The Problem So one kind of probability assignment is a counting assignment: P(') = a1 · T1(') + ··· + an · Tn(') wherein a1 + ··· + an = 1 and T1;:::; Tn are in [K], the space of complete consistent extensions of our background knowledge K. The philosophical concern relates to true arithmetic Th(N), the set of all sentences true on the standard model of arithmetic. This dilemma suggests that we should avoid appeal to counting assignments . But how can we do that? I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions The Problem So one kind of probability assignment is a counting assignment: P(') = a1 · T1(') + ··· + an · Tn(') wherein a1 + ··· + an = 1 and T1;:::; Tn are in [K], the space of complete consistent extensions of our background knowledge K. The philosophical concern relates to true arithmetic Th(N), the set of all sentences true on the standard model of arithmetic. If T1;:::; Tn included true arithmetic Th(N), then one would want to know what it is about our relationship to true arithmetic that grants it this preferred status. While if not, then one would want to know what about T1;:::; Tn make them reliable indicators of truth. Why these and not others? I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions The Problem So one kind of probability assignment is a counting assignment: P(') = a1 · T1(') + ··· + an · Tn(') wherein a1 + ··· + an = 1 and T1;:::; Tn are in [K], the space of complete consistent extensions of our background knowledge K. The philosophical concern relates to true arithmetic Th(N), the set of all sentences true on the standard model of arithmetic. If T1;:::; Tn included true arithmetic Th(N), then one would want to know what it is about our relationship to true arithmetic that grants it this preferred status. While if not, then one would want to know what about T1;:::; Tn make them reliable indicators of truth. Why these and not others? . This dilemma suggests that we should avoid appeal to counting assignments . But how can we do that? I. Intro II. Avoiding Counting Assignments III. Avoiding Alignment IV. Summary + Questions Counting Assignments have Atoms A counting assignment P induces a probability measure Pb : Borel([K]) ! [0; 1] on the Borel subsets of the space [K] by: Pb([']) = a1 · T1(') + ··· + an · Tn(') wherein ['] is the clopen set fT 2 [K]s : T j= 'g.